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Order of Operations, LCM, GCF & Fractions for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about arithmetic foundations in one place: order of operations (PEMDAS), greatest common factor, least common multiple, and every fraction operation — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupOrder of operations (PEMDAS)
The order of operations is the fixed sequence used to evaluate any expression with more than one operation. Getting this order wrong is one of the most common sources of careless errors on the PSAT/NMSQT, especially in longer expressions.
Evaluate: 6 + 2 · (5 − 3)² ÷ 4
Multiplication does not always come before division, and addition does not always come before subtraction — each pair is evaluated together, left to right, in the order it appears. Treating M as always before D (or A always before S) causes wrong answers on mixed expressions.
Greatest common factor (GCF)
The GCF of two or more numbers is the largest number that divides evenly into all of them. It's most often used to simplify fractions and to factor expressions.
Find the GCF of 36 and 48.
Prime factorization is the most reliable way to find a GCF for larger numbers — take the lowest power of every prime factor that appears in both numbers.
Least common multiple (LCM)
The LCM of two or more numbers is the smallest number that all of them divide into evenly. It's most often used to find a common denominator when adding or subtracting fractions.
Find the LCM of 8 and 12.
A quick shortcut connects GCF and LCM: GCF × LCM = the product of the two numbers. If you've already found one, you can solve for the other with a single division instead of factoring again.
Fraction operations
| Operation | Method |
|---|---|
| Add / Subtract | Find a common denominator (use the LCM), then add/subtract numerators |
| Multiply | Multiply numerators together and denominators together directly |
| Divide | Multiply by the reciprocal of the second fraction |
Add: 3/8 + 5/12
Divide: 2/5 ÷ 3/7
A very common error: forgetting to flip the second fraction before multiplying in a division problem. "Keep, change, flip" applies only to the second fraction — the first fraction never changes.
Simplifying & mixed numbers
Always reduce a fraction to lowest terms by dividing the numerator and denominator by their GCF. Improper fractions (numerator larger than denominator) can be converted to mixed numbers, and vice versa.
Convert 29/6 to a mixed number.
Before choosing an answer on a multiple-choice question, check whether your fraction is fully simplified — an unsimplified equivalent fraction may not match any of the listed answer choices even though it's mathematically correct.
GCF & LCM with fractions
GCF and LCM aren't standalone topics on the PSAT — they're the tools that make fraction arithmetic efficient. Use GCF to simplify a fraction to lowest terms, and LCM to find the smallest possible common denominator when adding or subtracting.
Simplify 42/56 to lowest terms.
Common PSAT traps
- Wrong order in mixed operations: multiplication/division and addition/subtraction are each done left to right as pairs, not in strict PEMDAS letter order.
- Confusing GCF and LCM: GCF is always smaller than or equal to both numbers; LCM is always larger than or equal to both numbers.
- Adding fractions without a common denominator: numerators can only be added or subtracted once the denominators match.
- Leaving an answer unsimplified: always reduce a final fraction answer to lowest terms before matching it to answer choices.
Test day strategy for order of operations, LCM, GCF & fractions
| Question signal | Fastest approach |
|---|---|
| Multi-step numeric expression | Apply PEMDAS strictly, working left to right within each tier |
| "Greatest common factor" language | Use prime factorization; take the lowest shared powers |
| "Least common multiple" or common denominator needed | Use prime factorization; take the highest powers present |
| Adding or subtracting unlike fractions | Find the LCM of the denominators first |
| Dividing fractions | Multiply by the reciprocal of the second fraction only |
| Fraction answer choices given | Simplify your result fully before comparing to the choices |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Order of Operations, LCM, GCF & Fractions 1
PEMDAS, greatest common factor, and least common multiple.
Start quiz → Quiz 2Order of Operations, LCM, GCF & Fractions 2
Fraction operations, simplifying, and mixed numbers.
Start quiz →